Perturbation solution of the coupled Stokes-Darcy problem

Microfiltration of particles is modelled by the motion of particles embedded in a Stokes flow near a porous membrane in which Darcy equations apply. Stokes flow also applies on the other side of the membrane. A pressure gradient is applied across the membrane. Beavers and Joseph slip boundary condition applies along the membrane surfaces. This coupled Stokes-Darcy problem is solved by a perturbation method, considering that the particle size is much larger than the pores of the membrane. The formal asymptotic solution is developed in detail up to 3rd order. The method is applied to the example case of a spherical particle moving normal to a membrane. The solution, limited here to an impermeable slip surface (described from 3rd order expansion), uses as an intermediate step the boundary integral technique for Stokes flow near an impermeable surface with a no-slip boundary condition. Results of the perturbation solution are in good agreement with O'Neill and Bhatt analytical solution for this case.